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A353834
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Nonprime numbers whose prime indices have all equal run-sums.
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25
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1, 4, 8, 9, 12, 16, 25, 27, 32, 40, 49, 63, 64, 81, 112, 121, 125, 128, 144, 169, 243, 256, 289, 325, 343, 351, 352, 361, 512, 529, 625, 675, 729, 832, 841, 931, 961, 1008, 1024, 1331, 1369, 1539, 1600, 1681, 1728, 1849, 2048, 2176, 2187, 2197, 2209, 2401
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
4: {1,1}
8: {1,1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
25: {3,3}
27: {2,2,2}
32: {1,1,1,1,1}
40: {1,1,1,3}
49: {4,4}
63: {2,2,4}
64: {1,1,1,1,1,1}
81: {2,2,2,2}
112: {1,1,1,1,4}
121: {5,5}
125: {3,3,3}
128: {1,1,1,1,1,1,1}
For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).
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MATHEMATICA
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Select[Range[100], !PrimeQ[#]&&SameQ@@Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]*k]&]
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PROG
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(Python)
from itertools import count, islice
from sympy import factorint, primepi
def A353848_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n: n == 1 or (sum((f:=factorint(n)).values()) > 1 and len(set(primepi(p)*e for p, e in f.items())) <= 1), count(max(startvalue, 1)))
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CROSSREFS
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These partitions are counted by A304442(n) - 1.
These are the nonprime positions of prime powers in A353832.
Including the primes gives A353833.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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